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Multiphase PMSG

8

2 Multiphase PMSG

In this chapter dynamic modeling of six phase permanent magnet synchronous machines are discussed. The physical parameters of the machine in stationary stator reference frame are given. Next, the modeling of the machine in synchronous rotating reference frame is studied. The two phase component transformation or vector space decomposition modeling of the machine is presented.

2.1 Introduction

The permanent magnet synchronous machine is a regular synchronous machine where the DCexcitation is replaced by permanent magnets. Excitation losses may reach up to 5% in small

machines [18].Having permanent magnets in the rotor circuit. These losses in the rotor are

eliminated. The PMSG has a smaller physical size, higher reliability and power density per

volume ratio. Because of these advantages, the PMSG are becoming an interesting solution

for wind turbine applications. However, the drawback of the PMSGs is their lack of voltage

control because of their constant excitation. Therefore, there have to be converters to support

voltage control for PMSG. But for variable speed wind turbine operation it is inevitable to use

converters for grid integration and have already been a standard.

Based on the rotor construction, PMSG can be either surface mounted permanent magnet

synchronous generators SPMSG or interior mounted permanent magnet synchronous

generators IPMSG. Interior mounted rotor is built with the magnets inserted in cavities in the

rotor structure. This creates saliency which results in reluctance torque component.

Based on the magnetic flux orientation, PMSG can be group as a radial flux and an axial flux

machine. Radial flux machine is constructed by placing stator around the rotor in such a way

to produce radial directed flux. In axial flux machine, the stator and the rotor are placed such

that the air gap is perpendicular to the direction of rotation there by the main flux crosses the

air gap in axial direction of rotation. Only Radial flux PMSG is studied here.

Based on the arrangement of stator windings, PMSG can also be divided as distributed

winding and concentrated winding. The generator that is available in the laboratory is

concentrated winding machine, therefore the modeling and control focus on this type of

machine.



Multiphase PMSG

Based on the number of phases, machines in general can be two phase, three phase or

multiphase (greater than three). Three phase machines have been studied for long time and

are widely available in numerous applications.

Multi phase machines provides several advantages such as

o Reduced amplitude and increased frequency of pulsating torque

o Lower current per phase for the same rated voltage

o Lower dc-link current harmonics converter fed system

o Higher reliability and

o Higher degree of freedom

The six phase synchronous machine has two identical stator windings which are assumed balanced star connected. Commonly, the two sets of winding can have a phase shift of 0, 30

and 60 degrees. Zero degree phase shift is exactly similar to three phase system. Sixty degree

phase shift forms symmetrical arrangement and can be reduced to three phase system because

two phases of different stars are always collinear. Thirty degree phase shift forms

unsymmetrical arrangement which cannot be further simplified. The thirty degree phase shift

arrangement is optimal with respect to voltage harmonic distortion and torque pulsation [7].

Therefore, thirty degree phase shift between star connections has been preference

arrangement and also in this project. But, the actual machine used for implementation in the

laboratory has 33.2725 degree separation between the two three phase groups.

2.2 Modeling Assumptions and Physical Parameters

In developing the mathematical model the following assumptions are made:

o The set of three-phase stator winding is symmetrical.

o The capacitance of all the windings can be neglected.o Each of the distributed windings may be represented by a concentrated winding.

o The change in the inductance of the stator windings due to rotor position is sinusoidal

and does not contain higher harmonics.

o Hysteresis loss and eddy current losses are ignored.

o The magnetic circuits are linear (not saturated) and the inductance values do not

depend on the current.



Multiphase PMSG

The arrangement of the windings of the machine is represented by their axis of flux lines as

shown in Figure 2-1.

Figure 2‐1 Six‐phase machine windings axis of flux lines

Mathematical model of six phase machines, like any other machine, can be started from the

basic the stator voltage equation which is given in matrix form as

[

(2.1)

Where, , ,

are stator voltage, stator current and stator flux linkages of machine respectively, and the

stator resistances are given

For magnetically coupled stator windings, the stator linkage flux is the sum of all fluxes

around it. This is represented by the following matrix equation:

(2.2)

is the flux linkage of the Permanent Magnet (PM) which links the stator windings can be

written as:

23 23 6 56 32 2.2.1



Multiphase PMSG

Where the permanent magnet flux linkage, and is the angle between magnetic axis of

phase and rotor as shown in Figure 2-1.

The inductance matrix includes the self inductance and the mutual inductances of all the

stator windings, and is given by

(2.3)

In the case of salient pole machine, the self-inductance of each stator phase winding willreach a maximum value whenever the rotor d-axis aligns with the axis of the phase winding

because the reluctance of the linkage flux path is minimum. This minimum reluctance

condition occurs twice during each rotation of the rotor so that the stator self-inductances are

represented by a constant component and a single periodic component, the higher harmonics

neglected [19]. As of the form

2, , , , , ,

, 23 23

6, 6 6 2.3.1

Where,

Lis leakage inductance a of stator winding.

Lis the constant component of the magnetizing inductance of a stator winding.

Lis amplitudes of the second harmonics of the magnetizing inductance.

As each of the stator windings is shifted in space relative to the others by 120 o the mutual

inductance between each of the stator windings is negative. When the rotor d-axis is midway

between axes of two of the windings, the magnitude of the inductance is maximum, [19].



Multiphase PMSG

Therefore, ignoring the mutual leakage inductance, the mutual inductance within the phase

group can be written as

12 23

12 22

2

12 22 12 26

2

2.3.2

The effect of mutual leakage inductance of the two groups of three phase windings voltageharmonic distortion and torque pulsation is appeared to be negligible when the separation of

the two set of windings is 30 o,[12] Therefore, the mutual coupling inductances between the

two groups can be written

√32 212

√32 2512

0 2

0 2512

√32 24

2.3.3



Multiphase PMSG

Six phase machine variables in stationary reference frame can be obtained using Clark Transformation and Modified Clark Transformation with respect 1, 1, 1and2, 2, 2respectively as seen in Figure 2-2.

Figure 2‐2 Dual Stationary Reference Frame

The stator voltage equation which is by equation Equations (2.1) is split in half, and Clarke

and inverse Clarke transformations on voltage, stator current and stator flux linkages of phase

group 1 , , 1 , and modified Clarke and inverse modified Clarke transformations

on voltage, stator current and stator flux linkages of phase group 2 , , 2. Then

the equivalent stator linkage flux in stationary frames is

2

(2.5)

The voltage equations in stationary frame become

sin

1 1 1 cos

sin

2 2 2 cos

(2.6)



Multiphase PMSG

The voltage equations, Equation (2.5), are dependent on the rotor position. In order to

eliminate the rotor position dependence and time varying equation, synchronous rotating

frame transformation is employed. Synchronous rotating frame is also called rotor reference

frame.

Figure 2‐3 Stator windings in synchronous rotating reference frame

Appling Angle Transformation on Equations (2.4) and (2.5), then the corresponding model

equivalent model of the machine on synchronous rotating frame

(2.7)

The voltage equations then

(2.8)

The equivalent circuit



Multiphase PMSG

(a) (b)

Figure 2‐4 Equitant circuit of six phase PMSG, (a) q axis (b) d axis

Electromagnetic Torque

The electromagnetic torque is the most important output variable that determines themechanical dynamics of the machine such as the rotor position and speed.

The electro-magnetic torque expression can be calculated from the air gap power . The air

gap power is the part of input power which does not contribute to resistive loss, ,or rate

of change of stored energy in the inductances. The rate of change of stored magnetic

energy could only be zero in steady state.

The total air gap flux contributes to the electromechanical torque,

, -electrical power

(2.9)

And if there are Ppoles, the electrical torque Mis

(2.10)

The toque expression given above is analogous to that of dc machine or that of three phase

AC machines in d-q plane.



Multiphase PMSG

2.4 Dynamic Modeling using Vector Space Decomposition

The vector Space decomposition theory has been introduced to transform the original six-

dimensional space of the machine into three orthogonal subspaces or planes. Vector

decomposition is based on the generalized two phase real component transformation of stator windings of n phase machine.

The generalized two-phase real component transformation is appropriate only for symmetrical

n-phase machines in which each stator phase is separated from each adjacent stator phase by

360 n⁄ electrical degrees [10]. As a result, this transformation cannot be applied directly to

the six-phase machine configuration with 30 degree separation. But, the generalized two-

phase transformation of a symmetrical 12 phase machine can be used to derive transformation

matrix for asymmetric six phase machine, see Figure 2-5.

Figure 2‐5 Winding axes for 12 phase and 6 phase windings.

For twelve phase machines, n=12, the transformation matrix [T12] is given in Appendix A1.

Then the twelve phase transformation matrix can be reduce to six phase transformation matrix

as [T6], it is voltage invariant form.

= 1 480 481 84 5 95 95 90 841 1 10 0 05 90 0 01 1 1

T T

(2.11)

Rearranging the above equations the voltage equations are



Multiphase PMSG

1

2

(2.12)

Equation (2.12) is has rotor position dependent voltage equations. Appling the voltage

equations referred to rotating space vector decomposition.

(2.13)

Equation (2.13) is similar to mathematical model of three phase machine in rotor reference

frame. It is used in vector control of six phase machine in similar approach to that of three

phase machines. Therefore vector space decomposition method of modeling multiphase

machines makes the control as ease as standard three phase machines.

Equivalent circuit in the three orthogonal planes can be derived using (2.12)-(2.13).



Multiphase PMSG

3P2 2 The toque expression given above is equivalent to that of DC machine or that of three phase

AC machines in d-q plane.

Therefore, systems of six phase machine can be simplified to two phase windings systems

which contribute for electromechanical energy conversion.

In general, multi phase machine modeling and control becomes easier using the vector

decomposition technique. All those derivations of equations are to understand the behavior of

multiphase machines which are used for control purpose. Before proceeding to control of six

phase machine, the next chapter presents about the converters used to operate the machine.



Converters

reduces ripple and or the harmonic distortion both in stator and dc link by creating smooth

transition in switching states. The advent of fast processing and advanced DSP modules

makes the real time control of complex algorithms easier and practical. So, it makes feasible

to think of the application of multi leg converters as one, instead.

Figure 3‐2 wind energy conversion six leg converter topology for six phase machines

As seen in Figure 3-2, the machine connected converters work in rectifier mode of operation

most of the time. Therefore, the basic principles and switching of three phase rectifiers are

discussed before jumping to six leg converters.

3.2 Operation of three phase rectifier

Three phase converter with fixed dc voltage polarity is called voltage source converter (VSC).

VSC can make a smooth transition from the inverter mode to rectifier mode by reversing the

direction of dc current; unlike current source rectifiers which reverse the dc voltage polarity.

To elaboration the basic operation of VSC converter, a single leg of converter is depicted in

below

Figure 3‐3 Single leg converter operation

Under Rectifier mode operation, the converter works like Boost converter. The upper switch

and lower diode work complementarily, the blue line shows the direction of current flow.

Under Inverter mode of operation, the converter works like Buck converter. The lower switch

and the upper switch and lower diode work complementarily which is shown by red line in



Converters

Figure 3-3. The relationship between input voltage (assumed sinusoidal) and converter

voltage, ignoring the harmonics, can be easily shown by a phasor

(3.1)

Figure 3‐4 Rectifier mode (left) and Inverter mode (right) of operation at unity power factor

From Figure 3-4, the converter voltage is higher that the input voltage in both modes near

unity power factor operation. Therefore, the dc link voltage should be sufficiently larger than

diode rectification voltage otherwise the converter will behave like common diode

bridge, . The active and reactive power flow is also well explained in [21]. When

three separate single legs are connected together, they form three phase converter called

voltage source converter.

There are many different switching techniques to control converters switching device. The

sinusoidal PWM and Space Vector PWM are discussed here, since they are the one used in

the implementation.

3.3 VSC -Sinusoidal Pulse Width Modulation

The Pulse Width Modulation is a technique of switching inverter power devices ON and OFF

at a carrier frequency in order to generate sequence of pulses whose average value forms the

reference signal. In another word, the reference signal is used to shape and control the

magnitude and frequency of converter output voltage when the reference signal is sinusoidal

waveform; the modulation is called sinusoidal PWM. In PWM, amplitude ratio or modulation

index is defined as the ratio of the peak amplitude of the modulating signal and the

amplitude of the triangular . The ratio of the frequency of carrier triangle and the

frequency of fundamental component PWM pulse pattern is called frequency modulation

ratio .

(3.2)



Converters

For 1, the modulation is called linear modulation in which the fundamental frequency

component of the converter voltage linearly varies with . For 1, the modulation

becomes nonlinear and is called over modulation.

3.3.1. Switching model In a three phase inverter only one of the switch in each leg is ON at a time; the upper three

switches can represent the converter using a the switching function is defined

1, 0, , (3.3)

The converter leg output voltage is given by

(3.4)

Using single carrier and three sinusoidal signals shifted by 120 o, as shown in Figure 3-6(a),

the sinusoidal PWM generated signal of phase and line voltage at the converter is shown

Figure 3-6 (b); the converter leg voltage has a dc value of .

Figure 3‐5 (a) Three phase PWM generation (b) phase and line voltages

3.3.2. Average value model of Converter leg

The average leg voltage at output can be obtained by integrating the switching voltage (3.4)

over a switching period and dividing by Ts.

,j= A, B, C (3.5)



Converters

where is the average line current and

Combining (3.9) - (3.11), the average model of three phase converter is Figure 3-7. Average

models make simulation fast though losing some degree of accuracy.

Figure 3‐6 Three phase VSC average model

3.4 VSC-Space Vector Pulse Width Modulation

Space vector Pulse Width Modulation (SVPWM) has interesting characteristics over

sinusoidal PWM. It improves the DC link voltage utilization and also reduces harmonics in

the voltages and currents. At any instant of time the vector sum of phase voltages in space

gives one vector –space vector. The use of the magnitude and frequency of this vector to

control switching states of the converter is named space vector modulation.

The switch mode converter connected to Machine is shown below

Figure 3‐7 Three phase converter connected to machine

I ∑d , , & . (3.11)



Converters

The maximum value of the average reference that can be used without distortion V is when

it is equal to the radius the circle inscribed hexagon. Then the peak value of the phase and its

root mean square can be calculated as

V Vcos30√ V

V √ V

V, V √ V 0.707V

(3.16)

This shows that using space vector modulation has 15 percent higher dc link voltage

utilization than that of sinusoidal modulation, compare with (3.10).

Next stage on SVPWM is identifying the sector where the reference voltage is laid. Then

calculate the time that the adjacent vectors inscribing the reference so as to keep the voltage-

second balance.

Figure 3‐9 synthesis of required voltage in sector 1

Depending on the criteria required zero vectors has to be selected to make the switching

frequency constant. When reference vector is in sector 1, see Fig.3.11, the duty cycle for the

adjacent vector V1 and V2, and zero vectors can be calculated as

d√ V sin α

d V √ V √ 3 cosα sinα

d 1 d d

(3.17)



Converters

The above formula can be used as general formula for the other sectors by adjusting the angle

the respective vectors. The detailed analysis for selection of zero vectors and the different

space vector algorithms are covered in [24]-[25].

Space vector source code is included in Appendix F.

3.5 Six Leg Converters

3.5.1. Model

Sig leg converter can be built, in similar fashion to that of three phases, by connecting six

bridge legs; or simply by connecting two VSC with common dc link voltage.

In this project, two identical three phase converters modules are used Figure 3.10(right side).

Figure 3‐10 Six leg converter

The six-phase voltage source converter contains a switching network of 12 power switches

arranged to form 6 legs, each leg connected to one phase of the generator. Only one of the

power switches of the same leg can operate in the ON state to avoid the short circuit of the dc

link so that the upper six switches determine the switching state of the converter. Thus, 64

possible states are available and having many switches states gives freedom in the control of

the converter and also it reduces the harmonics in voltage and current that would have been

introduced by having less number of switching states.

The neutrals of the generator are isolated from each other and from the converter. For a

balanced system, the voltage between the converter and machine for each set of three phase

windings becomes one third of the sum of the respective leg voltages, as shown in (3. 14) for

three phase machine

Therefore, using similar analysis the phase voltages can be written as



Converters

= 2 1 11 2 11 1 22 1 1

1 2 11 1 2

(3.18)

The converter leg voltage is product of the dc link voltage Vand the switching state of the

respective leg. For the six legs the converter voltages in matrix form can be written as

VN V SSSSSS=

V[

S (3.19)

Finally, substituting (3.19) to (3.18), the stator voltages of the generator in terms of the

switching states of the converter are

V VC[S, j a,b,c,x,y,z (3.20)

As shown in (3.21), by changing the switches states of the converter the stator voltages can be

controlled, or vice verse. The generation of control signal to the converter varies with

modulation scheme used. To take advantage of the higher number of switching states of thevector decomposition space vector PWM is used.

3.5.2. Sinusoidal Modulation

The six leg converter can be treated as two independent three phase system. For a balanced

load, the two converters will have similar reference signal which are separated by the phase

difference angle of the phase groups. DC link voltage utilization is as same as to that of three

phase converter. This modulation is very simple, in this project it is used in the initial stage of

the development of drive controller.

3.5.3. Space Vector Modulation-Vector Classification

Similarly, standard three phase space vector modulation can be use for the two converters.

First case is, using separate reference signals for each converter but the same switching states.

The same space vector modulation can be used, only by changing the input signal to the

function. This modulation strategy is implemented in this project.



Converters

Figure 3‐11 space vector modulation with different reference and same state vector position

The second case, is using the same reference signal but different position of switching states,

the switching states of one of the converter are uniformly displaced by the angle which is

equal to the phase difference of the two groups of three phase winding of the generator.

α

β

Vref

α

(2 / 3, 0)(‐2/3,0)

Figure 3‐12 space vector modulation with single reference and different vector position

3.5.4. Modulation - Vector Space Decomposition Space Vector PWM

As introduced earlier, for three phase converter modulation, space vector modulation is based

on the projection of the stator space vector in to stationary reference frame(s). From the

general two phase real component transformation, six space phase voltages can equally be

represented by vectors of three orthogonal stationary reference frames.

The stator voltages on three orthogonal planes can be found by multiplying (3.20) using six

phase transformation, Appendix A.

V VTC[

S (3.21)



Converters

Vα Vβ

VVVV=

√

1 0√ √

1

√ √ 01√ √ 00 √ √ 0 0 00 0 0 10 0 00 0 0

SS S SSS

Where: V VαVβ VVVV

As it is seen, (3.21), no voltage components are generated in the (z1-z2) subspace; this is

because of the two neutral of the machine are separated and system is assumed balanced.

There are 64 switching modes in each α

,β) and

x,y.The vector space diagram of the

switching states normalized with Vdc are shown below.

(a) (b)

Figure 3‐13 (a) projection of stator voltage (a,b) plane , (b)projection of stator voltage (x,y) plane, null voltage vectors: 0,63,7,56

Looking at the magnitude of the voltage vectors of above Figure 3-13, there are five vector

groups; the zero voltage group has 4 states Uo=0, the smallest voltage U1=0.299Vd group(12

states), the small-medium vector group has 12 states U2=0.577Vd, the medium voltage group

has 24 states U3=0.816Vd , and the largest voltage group has 12 states U4=1.115Vd;

From the two projection planes voltage diagrams, it is evident that the largest vectors (U4)

in α,β) plane are mapped to the smallest vectors in (x,y) plane. While the others two medium

voltages are same in both planes. So, does all the 128 vectors participated in

modulation/controlling of converter? As it is discussed in chapter 2, the space vector



Converters

decomposition modeling of generator shows only α,β ) planes contribute for

electromechanical energy conversion while x,y) corresponds to stator losses. That means, it

is wise to choose vectors which have largest projection on the α,β) and at the same time

smallest in

x,y). As a result, the vector diagram is simplified to 12 vectors-U4 group in α

,β)

and their 12 projection vectors-U1 group in x,y) as shown in Figure 3-14.

There are two 12 sectors each separated by 30 o. SVPWM technique of three phase converters

with 6 sectors can be extended to that of six phase converters-two 12 sectors.

Since the control of α,β) and x,y) orthogonal planes of one another, this gives additional

degree of freedom in the control of machine- converter system, so that the losses in x,y)

plane is intended to be as low as possible, by selecting four active vectors to generate

reference space vector in α

,β) and zero volt-second in

x,y).

(a) (b)

Figure 3‐14 vector-U4 group, of stator voltage (a,b) plane(x,y) plane

Supposing the reference vector is sector 1, the four adjacent vectors 45, 41, 9 and 11 are used.

The important features of these four adjacent voltages is that they lie in different quadrants

of x,y) plane which helps cancellation.

There are four zero vectors for each sector to choose from. Here, only one zero vector for

each sector is used and is selected to minimize the number of state changes of switch in each

leg. Table shows switching table for sector 1,

Table 3‐2 Switching table of sector 1

Sector 1

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

0.1

0.2

0.3

30

210

60

240

90

270

120

300

150

330

180 0



Converters

V0 63 1 1 1 1 1 1

V1 45 1 0 1 1 0 1

V2 41 1 0 1 0 0 1

V3 9 0 0 1 0 0 1

V4 11 0 0 1 0 1 1

As shown in Table 3.1, the all switch legs change state once in sampling time except leg b

which switches twice, and it is inevitable to have at least two state changes in sampling rate in

any combination. Therefore, it is possible to have different ways of choosing zero vectors and

thorough study can be done on the different combinations. The switching table for the other

sectors is included in Appendix C.

The dwelling time of these five vectors is calculated in such way that the voltage-second is

balanced [26],[27].Let T,T,T,Tand Tbe the dwelling time of the vectors

V,V,V,Vand Vrespectively. The voltage-second balance is

VT VTVTVTVT V T

T TTTT T

(3.22)

Further decomposing the vectors in to the subspaces which we have the information that the

voltage-second balance must be zero to minimize losses and is given by

VVVV1

VVVV VV1 VV1 VVVVVV1 VV1

TTTTT.TVTV00T (3.23)

where, Vis the projection of the i th voltage on the k-plane and Tis the dwelling time of the

vector Vover a sampling period T;Vα and Vβ are the projection of reference space vector

on α,β) plane .

Now, the timing signals of the legs or phases of the converter can be calculated by adding the

dwelling times according to switching states of the five vectors for the given sector. Table 3.3

shows the switching table for sector I.



Converters

Table 3‐3 switching table time and dwelling time of sector 1

Sector 1

V 63 1 1 1 1 1 1

T

T

T

T

T

T

V1 45 1 0 1 1 0 1

T 0 T T 0 T

V 41 1 0 1 0 0 1

T 0 T 0 0 T

V3 9 0 0 1 0 0 1

0 0

T 0 0

T

V4 11 0 0 1 0 1 1

0 0 T 0 T T

Ton T

T

+T

T T T TTTT

Note that: Tis the sampling time which has to be smaller than the switching time.

The ratio of the dwelling time to the sampling time can be defined as the duty cycle on that

sampling period,

d , i 0,1,2,3,4 (3.24)

The normalized switching gate signals then can be shown as

Figure 3‐15 switching gate signals



Converters

The sum normalized dwelling time of each vector gives the duty cycle of the switch or leg.

For this particular case, reference vector is in sector 1, the duty cycles of the six phases are as

follows.

d 1

d dd

d dd

d 1 d dd ddd(3.25)

The generation of switching signals must be handled carefully. The switching signals

generation should be handled in different way for the leg which changes state twice in a

sampling period.

The average model of six phase converter can be developed from the duty cycles of each

phase. The phase voltage is given by

V dV, j a,b,c,x,y,z (3.26)

Ignoring the witching losses the input and output are assumed to be equal, the DC link current

becomes as

∑ , , , , , ,

∑ , (3.27)

From (3.27) and (3.28), the average value model is composed of one controlled current source

on DC side and six controlled voltage sources from the AC side, see below



Converters

Figure 3‐16 Average value model of six leg converter

Using average value model, it is possible to do away with the gate signal generation processand at same time reducing the simulation time of the system.

To wrap up, the SVPWM has three main steps which have been discussed earlier: vector

selection, switching time calculation and gate signal generation. The Matlab codes for

selection of vector and switching time calculation are included in Appendix C.

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